Expected value of Wiener process

Question

I need to compute $E(W_1W_2W_3)$ where $W_t$ is a Wiener process.

When I try to derive it from the definition of expected value I get stuck with the integrals. Should I follow the propoerty $E(W_sW_t)=\min(s,t)$?

Answer 1

Set $X=W_1$ , $Y=W_2-W_1$ and $Z=W_3-W_2$. Note $X$ , $Y$ and $Z$ are independent and $$\mathbb{E}[X]=\mathbb{E}[Y]=\mathbb{E}[Z]=0$$ We have \begin{align*} \mathbb{E}[W_1W_2W_3]&=\mathbb{E}[X(X+Y)(X+Y+Z)]\\ &=\mathbb{E}[X^3+2X^2Y+XY^2+X^2Z+XYZ]\\ &=0 \end{align*} Remark $$\left\{ \begin{align} & \mathbb{E}\left[ {{W}^{2n+1}}(t) \right]=0\,\,\,\,\,\,\,\,\,\,\,\,\,\, \\ & \quad \mathbb{E}\left[ {{W}^{2n}}(t) \right]=\frac{(2n)!}{{{2}^{n}}n\,!}\,{{t}^{n}} \\ \end{align} \right.$$